ROTATIONAL DYNAMICS

Newton's 3rd Law – A Closer Look

● Consider 2 particles with action/reaction pair of forces:

– rd  = −  Newton's 3 Law → F 1 on 2 F 2 on 1 2 –  Puts no restrictions on the direction of F 1 on 2  r 12

● Symmetry Considerations 1 –  Attempt to write a formula for the force F 1 on 2 –  No “universal” xyz directions → What can F 1 on 2 depend on? –   Relative position vector r 12 and relative velocity vector v 12 only

●   If F 1 on 2 depends only on r 12 mathematically: –  Direction of r 12 → only “defined” direction in space   – F must point in direction of r (or opposite direction) 1 on 2 12 – Forces of this type are called “central” forces Central Forces and Torque 2  ●  × =  Mathematical definition of central force: r 12 F 1 on 2 0 r 12 – True in every reference frame! y 1

● Calculating in a particular reference frame S:  x    × r × F = 0 The quantity r i F i is called the “torque” 12 1 on 2  th ( i) on the i particle  −   ×  = r 2 r 1 F 1 on 2 0 Internal central forces produce zero net torque  ×   −  ×   = r 2 F 1 on 2 r 1 F 1 on 2 0 on a system  ×     ×   = r 2 F 1 on 2 r 1 F 2 on 1 0 External forces and non-central internal forces can exert non-zero net torque on a system ● Examples: – Central forces: gravity, electric – Non-central force: magnetic (depends on position and velocity)

Static Equilibrium

● “Equilibrium” → Fnet = 0 → aCM = 0

● “Static Equilibrium” → ai = 0 for every particle – Examples: buildings, bridges → (not perfectly static!) –  =  = ∑  ×   = Requires: F net , external 0 and net , external r i F i 0 i – Useful for calculating structural loads and stresses

● Examples: d – 3d/4 Shelf → calculate tension in cable m 2m – Calculate force of wall on plank d

– Door of width w and height h → draw direction of each Fhinge Angular Momentum ● Consider the net torque on a system of particles   d p  d    = ∑  ×   = ∑ × i   = ∑  r × p  net r i F i net r i net i i i i dt dt i

–  ×   th r i p i is referred to as “angular momentum”( L i ) of i particle – Internal, central forces exert zero net torque – Net torque must be provided by external forces:  d L  = total (Similar to Newton's 2nd Law ) net , external dt – If zero net external torque → angular momentum is conserved

  ● Both  and L depend on choice of origin – Unlike force and momentum (only depend on xyz directions) – However, the equation above is true in all reference frames Examples

● Pendulum at some instant (angle θ, speed v) θ d – Using top of string as origin: – Calculate torque and angular momentum m – Plug in to Tnet = dL/dt Repeat, using mass's lowest point as origin d/4

● Wooden board falls off table d – Mass m, starting from rest – Using edge of table as origin: – Calculate Tnet and aright edge of board at t=0 – (Assume board stays rigid → v proportional to r)

● Why does a helicopter need a tail rotor? CM Frame of a Physical System

● Empty space → homogeneous and isotropic – All points are physically the same → no point is “special” – All directions physically the same → no direction is “special” y y' y' x' y' vCM

x' x'

x ● System of particles → breaks this symmetry – A “special” reference frame can be defined for the system: – Origin = CM of system (called the “CM frame”) – x'y'z' axes → defined by particle positions (“principal axes”) –  Angular velocity vector CM → defined by particle velocities Angular Momentum in CM Frame

N  =  = 1 2  ∑ 1  2 psystem M system v CM KE system M system vCM mi v ' i 2 i = 1 2 ● For a system of particles measured in the CM frame: – Total momentum must be zero, but total KE can be non-zero!

● Angular Momentum (calculated in an inertial frame S):  = ∑ [ ×  ] = ∑ [    ×    ] Lsystem r i pi rCM r i ' mi vCM vi ' i i  = ∑ [ ×     ×     ×     ×  ] Lsystem mi r CM v CM r i ' v CM r CM v i ' r i ' vi ' i ∑  = ∑  = mi ri ' 0 mi vi ' 0 i i

 =   ∑  ×   Lsystem LCM mi r i ' v i ' “spin” angular momentum i

“orbital” angular momentum (zero in CM frame) Rotating CM Frame ● “Special” reference frame for a system of particles: – Must allow for rotating axes to account for angular momentum

● Let S be an inertial (non-rotating) CM frame – Let S' be a rotating CM frame with matrix R(t)   [    ] – Recall: v ' = R v −  × R r  = ∑  ×   =  Lsystem ' mi r i ' vi ' 0 For some “special” i  = ∑ [  ×   −  ×   ] = Ω Lsystem ' mi R r i R vi R r i 0 i  = ∑ [   ×   ] − ∑ [   ×  ×   ] = Lsystem ' mi R r i R v i mi R r i R r i 0 i i  = [ − ∑ [ ×  ×  ]] = Lsystem ' R Lspin , inertial mi r i r i 0 i   = ∑ [ ×  ×  ] This can be used to calculate the “special”  Lspin , inertial mi r i r i i for a given system (which makes L' = 0) Inertia Tensor  = ∑ [ ×  ×  ] Lspin , inertial mi r i r i i  = − ∑ [ ×  ×  ] Lspin , inertial mi r i r i i − −  0 z y 0 z y x  i i i i = − ∑  −   −   L m [ z 0 x z 0 x  ] spin , inertial i i i i i y i − − yi xi 0 yi xi 0  z  2  2 − −  y z x y z x x  i i i i i i   = ∑  −  2  2 −   ≡  L [ m  ] I spin , inertial i xi yi xi zi yi zi y i −z x − y z  x2  y 2  i i i i i i z

“Inertia Tensor” – fully describes the distribution of Reference frame for a system of mass in a system particles is almost complete:

Diagonal elements are called “moments of inertia” 1) origin → CM 2) angular velocity → using L and I Off-diagonal elements are called “products of inertia” 3) need to find “principal axes” Inertia Tensor for Mass Distributions  y 2  z2 −x y −z x  i i i i i i ● = ∑  −  2  2 −  For a system of particles: I mi xi yi xi zi yi zi i − −  2  2 zi xi yi zi xi yi

● Inertia tensor extends naturally to mass distributions:  2  2 − −  y z x y z x = ∫     −  2  2 −  I x , y , z dxdydz x y x z y z −z x − y z  x 2  y 2 ● Example: Calculate I for a flat disk in the xy-plane

M  2  2  x y R 1 2  2 M R 0 0   x , y , z  =   z  { R 4  = 1 2 z 0 elsewhere I  0 M R 0  4

1 2 0 0 M R 2 x y Principal Axes

 y 2  z2 −x y −z x I I I  i i i i i i xx xy xz = ∑  −  2  2 −  =   I mi xi yi xi zi yi zi I yx I yy I yz i − −  2  2  I I I zi xi yi zi xi yi zx zy zz ● Elements of inertia tensor depend on choice of xyz axes – Is there a “special” set of xyz axes for a given system? – Yes! Possible (but difficult) to find “principal axes” such that  L products of inertia are zero → in this reference frame: 

 2  2 =  y z 0 0 L x I xx x  i i = ∑   2  2  =  I mi 0 xi zi 0 L y I yy y i 0 0  x 2  y2 =    i i Lz I zz z Note: L and  can point in different ● For a system which is symmetric about an axis: directions!

– The symmetry axis is one of the principal axes of the system Rigid Body Rotation

● Physics definition of “rigid body” – System of particles which maintains its shape (no deformation) – i.e. velocity of particles in CM frame comes from rotation only – Notation: ω = angular velocity of rigid body (inertial CM frame) ω Fcent ● View from rotating CM frame Tension – Every particle stands still in equilibrium (I is constant) – Centrifugal forces balanced by internal stresses

● Rigid body model for solid objects → reasonably good – In reality, solids can 1) deform and 2) dissipate energy as heat – Rigid body model → not applicable to fluids, orbits, stars, etc. Example: Rotating Skew Rod ω y ● Rigid massless rod (length 2d): – Rotates as shown with mass m at each end φ x – At the instant shown: – Calculate Lsystem, using 2 different methods: –  ×  1) r i p i , and 2) calculate the inertia tensor – Notice L and ω don't point same direction!

● Is an external torque needed to sustain this motion? – If so, calculate it

● Use symmetry to guess the principal axes

– Verify guess by calculating inertia tensor in principal frame Fixed-Axis Rotation  d L ● ≠ 0 Common practice → hold ω constant (not L) → dt – If L and ω not parallel → “axle” must exert an external torque!

● Example: Rotating ceiling fan – If mass is symmetrically distributed: – L and ω parallel → fan turns smoothly – If mass distribution has asymmetry (poor alignment, etc.): – L and ω not parallel → fan wobbles

● High rotation speeds → wheels, lathes, etc. – Mounting must be able to exert external torque – Enough to handle a “tolerable” amount of asymmetry Parallel-Axis Theorem ● Often: fixed axis does not pass through CM of system – ω Example: door hinge – how to calculate Ldoor? –  =  ×       Recall Lsystem M total vCM rCM I principal –  =  ×  For rigid body rotation: v CM r CM d  =  ×   ×       Plug in: Lsystem M total r CM r CM I principal  =  ×  ×        Lsystem M total r CM r CM I principal  =          Lsystem I orbital I principal ● Parallel-Axis Theorem: – Let d be the perpendicular distance from CM to fixed axis = 2 =  2 I orbital , axis M d I axis I principal M d

=  1  2  2 = 1  2 I M 2 d M d M 2 d door about hinges 12 door door 3 door Rotational KE ● For a rotating rigid body (in inertial frame) :

= ∑ 1 2 = ∑ 1  ×  ⋅ ×   KE system mi vi mi r i r i i 2 i 2 = 1  ⋅ ∑  ×  ×   KE system [ mi r i r i ] (Using vector identity) 2 i

= 1 ⋅    = 1  ⋅   KE system I L (maximum angle between and L is 90º) 2 2 (For non-rigid bodies, must also include motion of particles toward/away from CM)

= 1 2  1 2  1 2 ● KE I I I In principal axis frame: principal 2 xx x 2 yy y 2 zz z

● Example: “average” frisbee toss – Estimate KEtranslation and KErotation in Joules Rolling Without Slipping

●   System's v CM and : independent of each other – No relation between translational / rotational motion in general – However, by using a force → the 2 can be coupled ω v ● Example: friction contact F – Drop a ball spinning at angular velocity ω on the floor – Relative velocity of ball's surface / floor causes kinetic friction – This force has 2 effects: –  1) pushes the ball to the right (affecting v CM )  – 2) exerts a CCW torque on the ball (affecting  )

● =  Rolling without slipping → condition where vCM R – vcontact=0 → fixed-axis rotation about contact pt. (static friction) Examples ● “Sweet spot” of baseball bat or tennis racket – Goal: Minimize effect of ball impact on hands – Assume quick impulse → Fhand has no effect – L If collision is elastic, calculate x such that: x – vend of bat = 0 (closest to hands) due to collision

● Belt around two rotors (m1, r1 and m2, r2) – Belt (mass mb) moves at speed v – Calculate the total KE of the system – What happens to belt as v gets large?

“Stability” of Rotating Objects ● Consider a rotating object which is deflected: – The larger Linitial gets: T dt – The smaller the deflection angle L

● Example: riding a bicycle – At low speed → leaning leads to falling over – At high speed → same amount of torque has less impact

● Coin (mass M, radius R) rolls without slipping on table – Traces out circular path (of radius d >> R) on table Video: – Coin must lean inward by (small) angle β to do so – Calculate β if circle takes time T to complete

Euler's Equations

●  =  Linear Momentum: p system m v CM and is conserved – Since m is a scalar and is constant → vCM is constant

 ● =   Angular Momentum: L spin , inertial I and is conserved – I → not a simple scalar and can vary with time – No such thing as “conservation of ω” → even with no torque! Chain rule:         d      =  × I   I    d Lspin , inertial d I   d  = =     I   dt dt dt dt (similar to F = m dv/dt ) To evaluate d I /dt, recall: In “rotating principal axis CM frame” → I' is constant for a rigid body  Notice  can change   “Euler's Equations” even if torque is zero! d I ' d I   ≡ 0 = −  × I dt dt y

Stable/Unstable Rotations ? ● Rigid-body rotation about any principal axis: ω x – Every particle in equilibrium (as viewed in rotating CM frame)  – Is this equilibrium stable? If has small off-axis component: – Do Euler's equations predict it will grow / shrink / neither?

●    ≪  Consider rotating principal-axis system with y , z x

– st   Euler's equations (to 1 order in y and z ) become: d  x = 2   −   −  I xx 0 (No external torque) d I I I I dt y − xx zz yy xx  2  = 0 2 x y dt I yy I zz  d “C” = Constant I y   I − I    = 0 yy dt xx zz x z C < 0 : rotation about x-axis is stable C > 0 : rotation about x-axis is unstable d  Largest & smallest moments of inertia: stable axis I z   I − I    = 0 Intermediate moment of inertia: unstable axis zz dt yy xx x y Precession ● In general → motion of ω very complicated – Euler's Equations – coupled nonlinear DE's – http://www.youtube.com/watch?v=GgVpOorcKqc – Terminology: “Precession” and “Nutation” – Precession → motion of ω simpler to describe than nutation

● Torque-free precession L – L has large stable-axis component → ω “circles” L – Example: spinning coin tossed in air “wobbles”

● Torque-induced precession – Causes L and ω to rotate around fixed axis

– Example: Precession of Earth axis due to Moon/Sun Gyroscopes ● Devices for studying / utilizing precession and nutation – Basic setup: single axis with low friction & good alignment – Can be used to produce strong L due to “spin”: – http://www.youtube.com/watch?v=hVKz9G3YXiw

● Commonly used in sensor systems – Can be used to measure orientation, latitude, acceleration, etc.

● Example: “Uniform Precession” – http://www.youtube.com/watch?v=8H98BgRzpOM&feature=related – Wheel has mass M, radius R, string attached distance d from center – Horizontal wheel spins at angular velocity ωspin, calculate ωprecession

“State” of a System

●   “Microscopic state” of a system of particles → mi , r i , vi – Given the system's current state and forces between particles: – Use Newton's Laws to predict future motion of each particle

● “Macroscopic state” of a system → Ptotal, KE, PE, L – Quantities associated with the system as a whole – Conservation Laws – predict the future motion of the system

● Experiments in early-mid 1900's – Showed that elementary particles (like e–) have L and energy – So they act more like systems than particles! – Quantum Mechanics explains how this can be possible